See for instance the proof of [1]}. For instance, it follows that \(\Theta _M(M)=M\) . We now turn to checking that \(\Theta _M\)  is a
lower density. If \(A,B \subseteq M\)  are such that \(A \ominus B \in {N}_{\mathrm {pu},k}\)  then \({H}^k(A \ominus B)=0\) , recall (iii). Consequently, \(\phi _M(A \cap \mathbf {B}(x,r)) = \phi _M(B \cap \mathbf {B}(x,r))\)  for all \(x \in \mathbb {R}^m\)  and \(r > 0\) . Thus, \(\Theta ^k_*(A,x) = \Theta ^k_*(B,x)\) . Since \(x\)  is arbitrary, \(\Theta _M(A) = \Theta _M(B)\) . This proves condition of REF . Condition (3) of REF  is trivial. In view of proving REF (3) we let \(A \in \overline{{B}(\mathbb {R}^m)}_M\) . According to condition (1) just proved and (REF ), there is no restriction to assume that \(A\)  is Borel. We ought to show that the equation \(\Theta ^k_*(\phi _M \hspace{2.5pt}{\vrule height7pt width.5pt depth0pt}\hspace{-0.2pt}\vbox {\hrule height.5pt width7pt depth0pt}\, A ,x) = \mathbb {1}_A(x)\)  holds for \({H}^k\) -almost every \(x \in M\) . Letting \(\psi = \phi _M \hspace{2.5pt}{\vrule height7pt width.5pt depth0pt}\hspace{-0.2pt}\vbox {\hrule height.5pt width7pt depth0pt}\, A\) , we infer from the Besicovitch Covering Theorem as in [2]} that \(\lim _{r \rightarrow 0^+} \frac{\psi (\mathbf {B}(x,r))}{\phi _M(\mathbf {B}(x,r))} = \mathbb {1}_A(x)\)  for \(\phi _M\) -almost every every \(x \in \mathbb {R}^n\) . In view of (REF ), it ensues that the sought
for equation holds \({H}^k\) -almost everywhere on \(M\) . To establish that \(\Theta _M\)  is a lower density, it remains to proves REF (4). Let \(A, B \in \overline{{B}(\mathbb {R}^m)}_M\) . We observe that \(\Theta _*^k(\phi _M \hspace{2.5pt}{\vrule height7pt width.5pt depth0pt}\hspace{-0.2pt}\vbox {\hrule height.5pt width7pt depth0pt}\, (A \cap B),x) \geqslant \Theta ^k_*(\phi _M,x) - \Theta ^{*\,k}(\phi _M \hspace{2.5pt}{\vrule height7pt width.5pt depth0pt}\hspace{-0.2pt}\vbox {\hrule height.5pt width7pt depth0pt}\, (M\setminus A),x) - \Theta ^{*\,k}(\phi _M \hspace{2.5pt}{\vrule height7pt width.5pt depth0pt}\hspace{-0.2pt}\vbox {\hrule height.5pt width7pt depth0pt}\, (M \setminus B),x)\) , for all \(x \in M\) . Now, as \(A\)  and \(B\)  are \(\phi _M\) -measurable, according to (REF ), if \(x \in \Theta _M(A) \cap \Theta _M(B)\) , then it follows from (REF ) that \( \Theta ^{*\,k}(\phi _M \hspace{2.5pt}{\vrule height7pt width.5pt depth0pt}\hspace{-0.2pt}\vbox {\hrule height.5pt width7pt depth0pt}\, (M \setminus A),x) = \Theta ^{*\,k}(\phi _M \hspace{2.5pt}{\vrule height7pt width.5pt depth0pt}\hspace{-0.2pt}\vbox {\hrule height.5pt width7pt depth0pt}\, (M \setminus B),x) = 0\)  and, in turn, referring to (REF ) again, that \(\Theta _*^k(\phi _M \hspace{2.5pt}{\vrule height7pt width.5pt depth0pt}\hspace{-0.2pt}\vbox {\hrule height.5pt width7pt depth0pt}\, (A \cap B),x)=1\) . Thus, \(x \in \Theta _M(A \cap B)\) . We have
shown that \(\Theta _M(A) \cap \Theta _M(B) \subseteq \Theta _M(A \cap B)\) . The
other inclusion is trivial, so that \(\Theta _M\)  is, indeed, a lower density.
